Quadratic Formula Calculator From Roots

This quadratic formula calculator helps you find the quadratic equation when you know its roots. Whether you're a student studying algebra or a professional working with quadratic functions, this tool provides a quick and accurate solution.

What is the Quadratic Formula?

The quadratic formula is a fundamental tool in algebra used to find the roots of a quadratic equation. A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and x represents the variable. The quadratic formula allows you to solve for x when you know the values of a, b, and c.

Note: The quadratic formula only works when a ≠ 0. If a = 0, the equation is no longer quadratic and should be solved using linear methods.

How to Find the Quadratic Equation from Roots

If you know the roots of a quadratic equation, you can find the original quadratic equation using the following steps:

Identify the roots of the quadratic equation (let's call them r₁ and r₂).
Use the factored form of the quadratic equation: (x - r₁)(x - r₂) = 0.
Expand the factored form to get the standard quadratic equation.

This process is essentially the reverse of finding the roots from a quadratic equation. Instead of starting with the quadratic equation and finding the roots, you start with the roots and find the equation.

The Formula

Given two roots r₁ and r₂ of a quadratic equation, the standard form of the quadratic equation can be written as:

x² - (r₁ + r₂)x + (r₁ × r₂) = 0

This formula is derived from the factored form of the quadratic equation. The sum of the roots (r₁ + r₂) gives the coefficient of the x term, and the product of the roots (r₁ × r₂) gives the constant term.

Worked Example

Let's find the quadratic equation given the roots 3 and -2.

Identify the roots: r₁ = 3, r₂ = -2.
Calculate the sum of the roots: 3 + (-2) = 1.
Calculate the product of the roots: 3 × (-2) = -6.
Write the quadratic equation using the formula: x² - (1)x + (-6) = 0, which simplifies to x² - x - 6 = 0.

Therefore, the quadratic equation with roots 3 and -2 is x² - x - 6 = 0.

Tip: You can verify your answer by plugging the roots back into the equation. For example, when x = 3: 3² - 3 - 6 = 9 - 3 - 6 = 0, which confirms that 3 is indeed a root.

Frequently Asked Questions

What is the difference between the quadratic formula and finding a quadratic equation from roots?

The quadratic formula is used to find the roots of a quadratic equation when you know the coefficients a, b, and c. Finding a quadratic equation from roots is the reverse process - you start with the roots and find the equation that would produce those roots.

Can I use this method for equations with complex roots?

Yes, this method works for complex roots as well. The sum and product of complex roots will result in complex coefficients in the quadratic equation.

What if I only have one root?

If you only have one root, you can use the fact that quadratic equations have two roots (real or complex). The other root would be the same as the given root, resulting in a perfect square quadratic equation.

Is there a way to find the quadratic equation if I don't know the roots?

If you don't know the roots, you would need to use other methods such as completing the square or using the quadratic formula to find the roots first, then use those roots to find the equation.